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3 Studia Mathematica

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Multilinear almost diagonal estimates and applications

100%

Bényi Á. , Tzirakis N.

2004

tom 164

nr 1

75-89

We prove that an almost diagonal condition on the (m + 1)-linear tensor associated to an m-linear operator implies boundedness of the operator on products of classical function spaces. We then provide applications to the study of certain singular integral operators.

Modulation space estimates for multilinear pseudodifferential operators

100%

Bényi Á. , Okoudjou K.

2006

tom 172

nr 2

169-180

We prove that for symbols in the modulation spaces $ℳ ^{p,q}$, p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

Anisotropic classes of homogeneous pseudodifferential symbols

100%

Bényi Á. , Bownik M.

tom 200

nr 1

41-66

We define homogeneous classes of x-dependent anisotropic symbols $Ṡ^{m}_{γ,δ}(A)$ in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in $Ṡ⁰_{1,1}(A)$ yield Calderón-Zygmund kernels, yet their L² boundedness fails. Finally, we prove boundedness results for the class $Ṡ^m_{1,1}(A)$ on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].